Rotating machines appear everywhere in our daily life, from high-tech applications, such as jet engines and computer hard disk drives (“HDD”) to ordinary household appliances, such as washing machines and refrigerator compressors. Rotating machines typically include a rotating part (rotor), a stationary part (stator or housing), and bearings. The rotors will have very different geometry for various applications. In jet engines, for example, the rotors are generally slender rods carrying relatively heavy disks with turbine blades. For an HDD, the rotor is generally a plurality of flexible disks attached in a spaced stack to a hub. Similarly, the stator or housing may have wide-varying geometry. The bearings connect or interface between the rotating and stationary parts. The bearings may include radial and thrust bearings to take the various loads during use. Typical bearing types include rolling element bearings, such as ball bearings, and hydrodynamic (fluid) bearings.
Traditional vibration analysis of rotating machines rests on several major assumptions. First, the rotors often have simplified geometry. For example, turbine engines are often modeled as rotating flexible rods with rigid or flexible disks. Conversely, HDDs are frequently modeled as rotating flexible disks on rigid hubs. As a result, vibration analysis models developed for one application often cannot be applied to other applications. Moreover, real rotors usually have much more complicated geometry than that utilized in existing models. Therefore, vibration analysis based on the simplified geometry often cannot predict responses accurately. Generally, the modeling of the housing and/or stator is minimal, to simplify the analysis. For example, the housing is often assumed to be rigid in modeling HDD systems. For jet engines, the housing is often modeled as a simple lumped-parameter system with a few degrees of freedom. Forced responses predicted from these simplified models may lose accuracy, even when the excitation frequency is low.
For the past decade, rapid technology advances in various industries call for more powerful vibration analysis tools for rotating machines. For example, high-speed, high-density HDDs require accurate geometric design of housing and rotors to meet stringent vibration specifications. For modem high-thrust turbine engines, housing flexibility can affect rotor performance considerably. All these new developments require greater vibrational analysis accuracy and challenge the assumptions made in traditional vibration analysis of rotating machines. Here is a specific example encountered in the HDD industry.
FIG. 1 shows a simplified model of an HDD spindle motor. The rotor consists of a rigid hub, two elastic disks (modeled by the classical plate theory), and a cantilever shaft (modeled as a Euler-Bernoulli beam). The stator is assumed to be rigid. The ball bearings are modeled as linear springs and dampers. In addition, the rotor spins with constant speed ω3. With these assumptions, the equations of motion have been derived using Lagrange's equation. FIG. 2 compares the theoretical natural frequencies of the rotor with experimental measurements as a function of the spin speed. Note that there is a significant discrepancy (about 15%) between the theoretical and experimental results for the first pair of modes denoted by (0,1), which are critical modes controlling read/write accuracy in hard disk drives.
The primary option to such vibrational analyses of an HDD spindle motor in spindle design is trial-and-error methods that are time-consuming and expensive. The time and expense limit the number of different rotor system designs that can be studied. As a result, the dynamic performance cannot be optimized with such few samples. Moreover, experimental evidence suggests that the vibrational response of the rotor may be significantly impacted by the flexible response of the fixed components, such as the base casting and the top cover. This implies that, with proper design of the fixed components, the rotor performance can be improved. However, existing disk/spindle vibration models do not include the flexibility of the fixed components and, therefore, are not effective for optimizing the dynamic response of the whole hard disk drive.